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## 3.2 An engineer's formulation

Slide : [ Weak form - By parts - time - space - Ax=b || VIDEO login]

After a section on what a physicist believes is a mathematician's view of the subject, it is time for an example. Using the format of a recipe'' that is applicable to a broad class of practical problems, we show here how the advection-diffusion equation (1.3.2#eq.2) is discretized using Galerkin linear finite elements (FEMs) and how it is implemented. Derive a weak variational form.
Multiply'' your equation by where the complex conjugate is necessary only if your equation(s) is (are) complex (1) Integrate by parts,
so as to avoid having to use quadratic basis functions to discretize the second order diffusion operator (2)

Assuming a periodic domain, the surface term has here been canceled, imposing natural boundary conditions. Discretize time
with a finite difference backward in time and using a partially implicit evaluation of the unknown where   (3) . All the unknowns have been reassembled on the left. Re-scale by , and Discretize space
using linear roof-tops and a Galerkin choice for the test functions (4)       (5)

Note how the condition is here used to create as many independent equations as there are unknowns . All the essential boundary conditions are then imposed by allowing and to overlap in the periodic domain. Since only the basis and test functions and perhaps the problem coefficients remain space dependent, the discretized equations are all written in terms of inner products, involving overlap integrals of the form . Reassembling them all in a matrix, Write a linear system
through which the unknown values from the next time step can implicitly be obtained from the current values by solving a linear system (6)

To relate the Galerkin linear FEM scheme (3.2#eq.5) with the code that has been implemented in the JBONE applet, it is necessary now to evaluate the overlap integrals. This is usually performed numerically using a quadrature. In the case of a homogeneous mesh , a constant advection and diffusion coefficient , the coming section shows how explicit expressions can be obtained from the same rules to define directly the matrix elements.

SYLLABUS  Previous: 3.1 Mathematical background  Up: 3 FINITE ELEMENT METHOD  Next: 3.3 Numerical quadrature